Double Integral of xyxy dxdy over the Area Bounded by yx^2 and yx

The evaluation of the double integral of the function xyxy over the region bounded by yx^2 and yx involves a step-by-step approach. This guide will walk you through the process, presenting the solution in a clear and detailed manner.

Step 1: Find the Points of Intersection

First, we need to find the points at which the curves yx^2 and yx intersect. We set the equations equal to each other:

x^2 x

By rearranging, we get:

x^2 - x 0, therefore xx - 1 0

Solving for x, we find:

x 0 and x 1

Step 2: Set Up the Double Integral

The region of integration is bounded by yx^2 below and yx above from x0 to x1. The double integral can be set up as follows:

∫01∫x^2x xyxy dy dx

Step 3: Evaluate the Inner Integral

We begin by evaluating the inner integral:

∫x^2x xyxy dy

This can be expanded:

xyxy x^2y - xy^2

We now compute the integral:

∫x^2x x^2y - xy^2 dy

Calculating each part separately:

For x^2y: ∫ x^2y dy x^2y^2/2

Evaluating from yx^2 to yx: x^2x^2/2 - x^2x^4/2 x^6/2 - x^4/2 (x^6 - x^4)/2

For xy^2: ∫ xy^2 dy xy^3/3

Evaluating from yx^2 to yx: x x^3/3 - x x^2^3/3 x^4/3 - x^7/3 (x^4 - x^7)/3

Combining both parts of the inner integral:

(x^6 - x^4)/2 - (x^4 - x^7)/3

Finding a common denominator which is 6:

(3x^6 - 3x^4 - 2x^4 2x^7)/6 (3x^6 - x^4 - 2x^7)/6

Step 4: Evaluate the Outer Integral

Now we evaluate the outer integral:

(1/6) ∫01 (3x^6 - x^4 - 2x^7) dx

Calculating each integral:

For 3x^6: ∫ 3x^6 dx 3x^7/7

Evaluating from x0 to x1:

3/7

For -x^4: ∫ -x^4 dx -x^5/5

Evaluating from x0 to x1:

-1/5

For -2x^7: ∫ -2x^7 dx -2x^8/8

Evaluating from x0 to x1:

-1/4

Combining these results:

3/7 - 1/4 - 1/5

Finding a common denominator which is 140:

(3·20 - 1·35 - 1·28)/140 (60 - 35 - 28)/140 -3/140

Step 5: Final Result

Multiplying by 1/6:

(1/6)·(-3/140) -3/840 -1/280

Therefore, the value of the double integral is:

boxed{-1/280}